Numerical Investigation of the Apparent Viscosity Dependence on Darcy Velocity During the Flow of Shear-Thinning Fluids in Porous Media

  • Antonio Rodríguez de CastroEmail author
  • Mehrez Agnaou


The viscosity exhibited by shear-thinning fluids within the interstices of a porous medium differs depending on pore dimensions and injection velocity. Therefore, predicting the macroscopic value of viscosity required as input to Darcy’s law is challenging and needs accurate identification of the characteristic microscopic dimensions dominating global pressure losses. The most common approach consists of defining an apparent “in situ” shear rate which can be used in the bulk constitutive equation of the fluid to predict viscosity during flow through the porous medium. The dependence of this apparent shear rate on Darcy velocity has traditionally been assumed to be linear, which is appropriate in the case of Newtonian fluids and power-law fluids. However, yield stress and plateau viscosities that can potentially affect such dependence are not captured by power-law model, so the linear assumption may lead to inaccurate viscosity predictions. For this reason, a set of two-dimensional (2D) flow problems were considered and solved numerically to assess the effects of the shear rheology model, the pore size distribution and the microstructural complexity on the value of the apparent shear rate. In order to facilitate the analysis, the microscopic features of all the investigated porous media were well characterized through pore network modelling. The present results prove the inability of traditional approaches to predict the macroscopic viscous pressure losses generated during the creeping flow of widely used Herschel–Bulkley and Carreau fluids. In particular, the existence of a yield stress or a plateau viscosity induces significant deviations from linearity in the relationship between apparent shear rate and Darcy velocity. The importance of these deviations is, in turn, shown to be highly affected by the dispersion of the pore size distribution and the degree of shear thinning. Moreover, the reasons for such observations are discussed using an analytical approach.


Shear-thinning fluids Yield stress Numerical simulations Apparent viscosity 2D porous media 

List of Symbols

Roman Letters


Consistency of a given power-law fluid (Pa sn)


Flow behaviour index of a given power-law fluid


Power-law index of a given Carreau fluid


Hydraulic aperture of a bundle of rectangular channels (m)


Aperture classes of the non-uniform channels in the bundle of rectangular channels model (m)


Consistency of a given Herschel–Bulkley fluid (Pa sn)


Intrinsic permeability of a given porous medium (m2)

\( m_{1} \), \( m_{2} \)

Means of the distributions in a weighted sum of two normal laws (μm)


Flow index of a given Herschel–Bulkley fluid


Number of (ui, \( \nabla P_{i} \)) data obtained in a given numerical experiment


Number of capillaries in a bundle of cylindrical capillaries


Probability of a given aperture class hi


Probability of a given radius class ri


Relative contribution of a given pore class ri to the total flow rate of a yield stress fluid at a given pressure gradient \( \nabla P_{j} \)


Probability in terms of volume of a given pore radius class in a porous medium


Relative contribution of a given pore class ri to the total flow rate of water at a given pressure gradient \( \nabla P_{\text{j}} \)


Pressure (Pa)

\( q\left( {\nabla P,\;r} \right) \)

Volumetric flow rate of a Herschel–Bulkley fluid through a capillary of radius \( r \) as a function of \( \nabla P \) (m3/s)


Radius of a cylindrical pore (m)


Pore radius classes in a bundle of cylindrical capillaries (m)


Smallest pore radius class in a bundle of cylindrical capillaries (m)


Largest pore radius class in a bundle of cylindrical capillaries (m)

\( \bar{r} \)

Hydraulic radius of a bundle of cylindrical capillaries (m)


Radius of the circular cross-sectional area of the bundle of cylindrical capillaries (m)


Darcy velocity (m/s)

Greek Letters

\( \alpha \)

Shift factor relating \( \dot{\gamma }_{\text{pm}} \) to u

\( \alpha_{\text{N}} \)

Shift factor for the injection of a Newtonian fluid

\( \dot{\gamma } \)

Shear rate (s−1)

\( \dot{\gamma }_{\text{pm}} \)

Apparent shear rate for the flow of a complex fluid through a porous medium (s−1)

\( \dot{\gamma }_{\text{w}} \)

Wall shear rate in a circular channel for the flow of a yield stress fluid (s−1)

\( \dot{\gamma }_{{{\text{w}},{\text{Newtonian}}}} \)

Wall shear rate in a circular channel for Newtonian flow (s−1)


Porosity of a given porous medium

\( \lambda \)

Time constant (also known as relaxation time) of a given Carreau fluid (s)

\( \mu \)

Shear viscosity (Pa s)

\( \mu_{0} \)

Lower Newtonian plateau viscosity of a given Carreau fluid (Pa s)

\( \mu_{\infty } \)

Upper Newtonian plateau viscosity of a given Carreau fluid (Pa s)

\( \mu_{ \rm max } \)

Maximum viscosity value allowed in the direct numerical computations

\( \mu_{\text{pm}} \)

Apparent “in situ” shear viscosity of a given non-Newtonian fluid (Pa s)

\( \sigma_{1} \), \( \sigma_{2} \)

Standard deviations of the distributions in a weighted sum of two normal laws (μm)

\( \tau \)

Shear stress (Pa)

\( \tau_{0} \)

Yield stress of a given Herschel–Bulkley fluid (Pa)

\( \tau_{\text{w}} \)

Wall shear stress (Pa)



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Arts et Métiers ParisTechChâlons-en-ChampagneFrance
  2. 2.Laboratoire MSMP – EA7350Châlons-en-ChampagneFrance
  3. 3.Department of Chemical EngineeringUniversity of WaterlooWaterlooCanada

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